# Properties

 Label 2888.2.a.e Level $2888$ Weight $2$ Character orbit 2888.a Self dual yes Analytic conductor $23.061$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,2,Mod(1,2888)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2888.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2888 = 2^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2888.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$23.0607961037$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} + 3 q^{5} - 2 q^{9}+O(q^{10})$$ q + q^3 + 3 * q^5 - 2 * q^9 $$q + q^{3} + 3 q^{5} - 2 q^{9} - 4 q^{11} - 5 q^{13} + 3 q^{15} - 5 q^{17} - q^{23} + 4 q^{25} - 5 q^{27} + 3 q^{29} + 4 q^{31} - 4 q^{33} + 2 q^{37} - 5 q^{39} - 5 q^{41} - 11 q^{43} - 6 q^{45} - 5 q^{47} - 7 q^{49} - 5 q^{51} - 9 q^{53} - 12 q^{55} + 13 q^{59} - q^{61} - 15 q^{65} - 5 q^{67} - q^{69} + q^{71} - 9 q^{73} + 4 q^{75} + 17 q^{79} + q^{81} + 16 q^{83} - 15 q^{85} + 3 q^{87} + 3 q^{89} + 4 q^{93} - 13 q^{97} + 8 q^{99}+O(q^{100})$$ q + q^3 + 3 * q^5 - 2 * q^9 - 4 * q^11 - 5 * q^13 + 3 * q^15 - 5 * q^17 - q^23 + 4 * q^25 - 5 * q^27 + 3 * q^29 + 4 * q^31 - 4 * q^33 + 2 * q^37 - 5 * q^39 - 5 * q^41 - 11 * q^43 - 6 * q^45 - 5 * q^47 - 7 * q^49 - 5 * q^51 - 9 * q^53 - 12 * q^55 + 13 * q^59 - q^61 - 15 * q^65 - 5 * q^67 - q^69 + q^71 - 9 * q^73 + 4 * q^75 + 17 * q^79 + q^81 + 16 * q^83 - 15 * q^85 + 3 * q^87 + 3 * q^89 + 4 * q^93 - 13 * q^97 + 8 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 3.00000 0 0 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.2.a.e 1
4.b odd 2 1 5776.2.a.h 1
19.b odd 2 1 2888.2.a.c 1
19.c even 3 2 152.2.i.a 2
57.h odd 6 2 1368.2.s.g 2
76.d even 2 1 5776.2.a.o 1
76.g odd 6 2 304.2.i.b 2
152.k odd 6 2 1216.2.i.e 2
152.p even 6 2 1216.2.i.i 2
228.m even 6 2 2736.2.s.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.a 2 19.c even 3 2
304.2.i.b 2 76.g odd 6 2
1216.2.i.e 2 152.k odd 6 2
1216.2.i.i 2 152.p even 6 2
1368.2.s.g 2 57.h odd 6 2
2736.2.s.q 2 228.m even 6 2
2888.2.a.c 1 19.b odd 2 1
2888.2.a.e 1 1.a even 1 1 trivial
5776.2.a.h 1 4.b odd 2 1
5776.2.a.o 1 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2888))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} - 3$$ T5 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 3$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 5$$
$17$ $$T + 5$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T - 3$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T + 5$$
$43$ $$T + 11$$
$47$ $$T + 5$$
$53$ $$T + 9$$
$59$ $$T - 13$$
$61$ $$T + 1$$
$67$ $$T + 5$$
$71$ $$T - 1$$
$73$ $$T + 9$$
$79$ $$T - 17$$
$83$ $$T - 16$$
$89$ $$T - 3$$
$97$ $$T + 13$$